-
539
0022-4715/02/0400-0539/0 2002 Plenum Publishing Corporation
Journal of Statistical Physics, Vol. 107, Nos. 1/2, April 2002 (
2002)
On Implementation of Boundary Conditions in theApplication of
Finite Volume Lattice BoltzmannMethod
Y. T. Chew,1 C. Shu,1 and Y. Peng1, 2
1Department ofMechanical EngineeringNational University of
Singapore, Singapore 119260.2 To whom correspondence should be
addressed; e-mail: [email protected]
Received February 13, 2001; accepted September 3, 2001
A new implementation of boundary condition based on the
half-covolume andbounce-back rule for the non-equilibrium
distribution function for the finitevolume LBM is proposed here.
The numerical simulation results for the expan-sion channel flow
and driven cavity problem indicate that this method is work-able
for arbitrary meshes. In addition, the fourth order RungeKutta
scheme isfound to be a practical way in the LBM to accelerate the
calculation speed.
KEY WORDS: Lattice-Boltzmann equation; finite volume; boundary
conditions;half covolume scheme; bounce back.
1. INTRODUCTION
In recent years, the LBM has been developed into an alternative
useful toolto solve complex fluid flows. Although it has notable
advantages over theconventional methods, there are still some
limitations in the implementa-tion of LBM models. One of these is
that the LBM scheme on a Cartesian-like grid is restricted to a
special class of uniform and regular spatial lattices.Some workers
have attempted to extend the applicability to the
irregularlattices. Succi (1, 2) was the first to propose a
finite-volume formulation of theLBM. However, the empirical
formulae used are quite complicated evenfor the simple rectangular
meshes, and a free parameter has to be intro-duced and adjusted in
order to minimize the numerical diffusion. Toimprove it, Chen (3)
developed another finite-volume scheme. With properlychosen forms
of the state-flux functions, both exact conservation laws
andequilibrium balance conditions are achieved as in the original
LBM. In the
-
new method proposed by He, Luo and Dembo, (4) an interpolation
step isintroduced after stream and collision steps to determine the
density distri-butions at the grid points for the next time step.
Filippova and Hanel (5)
presented the multiscale LBM scheme with the boundary-fitting
formula-tion on the curvilinear boundaries. It uses the concept of
hierarchical gridrefinement. The calculation is based on a coarse
grid covering the wholeintegration domain. In a critical region, a
finer grid is superposed to thebasic grid. The calculation proceeds
with large time-step according tothe coarse grid; while on the
finer grids, several time steps according to therefinement ratio
are performed to advance to the same time level. Thisfeature is
very important for the computations of time-dependent flows.For the
computation of steady-state incompressible flows, the use ofseveral
smaller time-steps on the fine grid will increase the
computationaltime. In order to remove this drawback, the use of
smaller amount of timesteps on the fine grid is proposed. For the
time-dependent computation,this is connected with the change of
molecular speeds (6) on the fine gridso that the temporal accuracy
will not be impaired in certain limits. Forsteady-state
computation, the saving of CPU time can be even larger, sincethe
same amount of time steps can be chosen on coarse and fine grids
aswell. Recently, a new method is proposed by Xi, (710) which can
be used onirregular meshes with arbitrary connectivity. It is based
on modern finite-volume methods (11) and keeps the simplicity of
the conventional LBM.
The main focus associated with this new scheme is the
implementationof the boundary conditions. The half-covolume
technique is introduced andused by Peng and Xi (12) at the solid
boundary. This method is quite generaland simple in the sense that
it does not assume the fluid properties and theorientations of the
boundary walls. It is very robust when the inlet andoutlet
boundaries are periodical. However, it will cause some problemswhen
it is used in other flow problems such as the velocity profile
beinggiven at the inlet. In order to solve this problem, a new
implementation ofboundary condition based on the half-covolume and
bounce-back rule forthe non-equilibrium distribution function for
the finite volume LBM isproposed here. Besides, this problem is
more severe at the corner points. Indriven cavity problem, the left
bottom small vortex cannot be obtainedwhen no special treatment is
used at the corner points. So some additionaltreatments are used at
the corner nodes. As will be shown in Section 3.2,the physical
background of bounce back rule is the compliance of
Gradsthirteen-moments. (13) When the compliance of Grads
thirteen-moments orthe bounce back rule is broken, the no-slip
boundary condition on the solidwall may not be guaranteed.
Considering the fact that both the completebounce back scheme (14)
and half-covolume technique do not distinguishamong distribution
functions, the combination of the half-covolume and
540 Chew et al.
-
the bounce back scheme are very easy to implement in a computer
code.The proposed approach is validated by its application to solve
the 2Dexpansion channel flow and the driven cavity flow.
2. THE FINITE-VOLUME LBM MODEL
The finite-volume approach starts with the lattice Boltzmann
equationin differential form, which reads:
fit +vi Nfi=Wi+avi F (1)
where
Wi=1y[fi(x, vi, t)f
eqi (x, vi, t)]
feqi =wir 51+3vi uc2 +9(vi u)2
2c23u2
2c26
vi=0 i=0(cos[(i1) p/2], sin[(i1) p/2]) c i=14`2 (cos[(i5)
p/2+p/4], sin[(i5) p/2+p/4]) c i=58
w0=4/9, wi=1/9 for i=14 and wi=1/36 for i=58
a=1;Civ2ix=1;C
iv2iy
Figure 1 shows a finite element surrounding an interior node P.
Here P, P1to P8 are the grid points. A to H represent the edges of
the control volumeover which integration of the PDE is performed.
A, C, E, and G are themidpoints of the edge PP1, PP3, PP5 and PP7
respectively. B, D, F, and Hare the geometric center of element
PP1P2P3, PP3P4P5, PP5P6P7 andPP7P8P1 respectively.
The cell-vertex type is used here. In this type of formulation,
all thedensity distribution functions at the grid nodes are known
while the distri-bution functions at other locations are
interpolated from the known valuesat the grid points using standard
interpolation technique.
On Implementation of Boundary Conditions 541
-
P3 P2 P4
P5
P6 P7
P8
P1
P E
A
B C D
F G H
Fig. 1. Diagram of a finite element surrounding an interior node
P.
The integration of the first term in Eq. (1) is approximated
as
FPABC
fit ds=
fi(P)t sPABC (2)
where sPABC is the area of PABC and fi(P) is the fi value at
grid point P.In what follows, the grid-node index is given in
parentheses following thefi values. In the above equation, an
approximation that fi is constant overthe area PABC is used to
prevent solving a set of equations.
The integration of the second term of Eq. (1) will give fluxes
throughthe four edges PA, AB, BC, and CP. Since the summation over
all thepolygons like PABC, PCDE, PEFG, and PGHA will be done, the
net fluxthrough internal edges (PA, PC, PE, PG) will cancel out.
Therefore, theexplicit expression for the internal edges will be
omitted. That is
FPABC
vi Nfi ds=vi FABfi dl+vi F
BCfi dl+Is (3)
where Is represents fluxes through internal edges. With the
standardassumption of bilinearity of fi in quadrilateral elements,
the flux is thengiven by
FPABC
vi Nfi ds=vi nABlAB[fi(A)+fi(B)]/2
=vi nBClBC[fi(B)+fi(C)]/2+Is (4)
where nAB and nBC are the unit vectors normal to the edge AB and
BC,respectively, and lAB and lBC are the lengths of AB and BC,
respectively.
542 Chew et al.
-
With the assumption of bi-linearity of fi and feqi over the
quadrila-
teral elements, the integration over the collision term of Eq.
(1) results inthe following formula:
FPABC
1y(fif
eqi ) ds=
sPABCy[Dfi(P)+Dfi(A)+Dfi(B)+Dfi(C)]/4
(5)
where
Dfi(P)=fi(P)feqi (P)
Dfi(A)=fi(A)feqi (A)
Dfi(B)=fi(B)feqi (B)
Dfi(C)=fi(C)feqi (C)
Here fi(A), fi(B), fi(C) and their corresponding equilibrium
particle dis-tribution functions feqi (A), f
eqi (B), f
eqi (C) are the values at non-grid nodes
A, B, and C, respectively. This may be obtained by interpolation
from thefour grid nodes at element PP1P2P3,
fi(A)=[fi(P)+fi(P1)]/2
fi(B)=[fi(P)+fi(P1)+fi(P2)+fi(P3)]/4
fi(C)=[fi(P)+fi(P3)]/2
feqi (A)=[feqi (P)+f
eqi (P1)]/2
feqi (B)=[feqi (P)+f
eqi (P1)+f
eqi (P2)+f
eqi (P3)]/4
feqi (C)=[feqi (P)+f
eqi (P3)]/2
With these results, the integration of Eq. (1) over the polygon
PABC iscomplete. The integration over the whole control volume
ABCDEFGH isjust the sum of contributions from all these terms over
different polygonsPABC, PCDE, PEFG, and PGHA. Therefore, fi at grid
node P is updatedas follows:
fi(P, t+Dt)=fi(P, t)+DtsP1 Caround P
(collisions) Caround P
(fluxes)2+avi F(6)
where sP is the total area of the control volume around grid
node P, andcollisions and fluxes refer, respectively, to the
finite-volume-integratedcontributions from the collision term and
fluxes.
On Implementation of Boundary Conditions 543
-
3. IMPLEMENTATION OF BOUNDARY CONDITIONS
3.1. Half-Covolume Scheme
Let P, P5, and P1 are boundary nodes separating the fluid (upper
half)from the lower half. As for the interior fluid nodes, the
value of fi at P isupdated through Eq. (6) by covolume integrals.
At the boundary, thecovolume is not complete in the 2p direction as
the polygons PEFG andPGHA are not included. This leads to the
difference when integrating thesecond term of Eq. (1) over polygons
PABC and PCDE. The flux termsover the edges PA and EP, which are
omitted in the case of interior nodes,must be included in the
calculation. They are actually easy to evaluate byEq. (4). The
velocity of the boundary wall is used when feqi for the bound-ary
nodes are calculated in order to enforce the no-slip boundary
condi-tion. This is an effective way to implement the boundary
condition for thefluid problems where the inlet and outlet are
periodic conditions. This canbe verified by the following
cases.
(1) Two-dimensional Poiseuille flow between two parallel
plates.
The external force is F=2.60410 5ex. The total 6432 mesh
pointsare used. The analytical solution for the case is
ux(y)=FL2/(8ru) N1(2y/L1)2M, where L is the channel width. Figure 2
shows the numerical resultswith the analytical solution. It can be
seen that the agreement is excellent.
(2) Two-dimensional rotating Couette flow between two
concentriccylinders.
0
0.002
0.004
0.006
0.008
0.01
0.012
0 5 10 15 20 25 30 35
y
ux(y)
present
analytic
Fig. 2. Numerical velocity profile for the Poiseuille flow as
compared with analytic solution.
544 Chew et al.
-
00.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
30 35 40 45 50 55 60
r
Vth
eta(r)
present
analytic
Fig. 3. Numerical velocity profile for rotating Couette flow as
comparedwith analytic solution.
The outer cylinder rotates with velocity Veh, while the inner
cylinder isstationary. In this simulation, the radii of the two
cylinders are R1=30 andR2=60, the velocity V=0.01. The 18030 mesh
points are used. Theanalytical solution for the problem is
uh(r)=(V2R2r
V2R21R2r )/(R
22R
21).
Figure 3 shows the numerical result of the steady velocity
profile and thecorresponding analytical solution. One can see from
Fig. 3 that the agree-ment is also excellent.
(3) Plane Couette flow with a half-cylinder of radius R resting
on thebottom plane.
The meshes are generated using elliptic grid generation method
asshown in Fig. 4. In this simulation, R=20 is for the radius of
cylinder,
Fig. 4. Meshes used for flow past a half-cylinder resting on a
plane.
On Implementation of Boundary Conditions 545
-
00.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
20 40 60 80 100 120 140
y
ux(0,y)
present
result of Xi
Fig. 5. Velocity field u in the center (x=0) of the channel for
flow past a half-cylinderresting on a plane.
U=0.1 is for the top plane speed, and the plane is 9.5R6R. The
meshsize used is 10060. Figure 5 shows the velocity profile across
y in thecenter of the channel. Good agreement between present
results and thosegiven by Xi (8) was found.
3.2. Half-Covolume plus Bounce Back Method
It was found from the above section that half covolume method is
veryeffective in solving flow problems where the inlet and outlet
are periodical.However, it would cause some problems when it is
used in other cases suchas the velocity profile being given at the
inlet. The reason for this may lie inthe inconsistency with Grads
13 moments expansion, which is needed to besatisfied for a robust
and efficient boundary condition.
According to Grads 13-moment system, the non-equilibrium
densitydistribution function can be written as
fneq=feq 1P : (viu)(viu)2pRT
S (viu)2pRT11 (viu)2
(D+2) RT22 (7)
where P and S are the stress tensor and heat flux vector,
respectively. Theterms involving O(u) and higher order can be
neglected since the non-equi-librium distribution itself is very
small, which leads to
fneq=feq 1P : vivi2(RT)2
Svi2(RT)211 v2i
(D+2) RT22 (8)
546 Chew et al.
-
7
1
5
8
2 6
3
4
Fig. 6. Schematic plot of velocity directions of the 2-D model
at the bottom wall.
For the isothermal flow, the heat transfer term can be
neglected, and thenit can be approximated by
fneq=feqP : vivi2(RT)2
(9)
So
fneq, isoa =fneq, isob (10)
where a and b have the opposite direction.On the solid wall, Eq.
(10) is actually the bounce back condition. This
implies that the physical background of bounce back rule is the
complianceof Grads thirteen-moments. In the following applications,
we combine thehalf-covolume and the bounce back rule for the
non-equilibrium distribu-tion to implement the boundary condition.
As an example, we consider thecase of the bottom wall. The nine-bit
model is shown in Fig. 6. For thebottom wall, the distribution at
direction 7, 4, and 8 are determined by half-covolume, and the
distribution at direction 5, 2, and 6 are determined bythe bounce
back rule for the non-equilibrium distribution through Eq.
(10).
To test the validity of this new implementation of the boundary
con-dition, the numerical simulations for the expansion channel
flow arecarried out. This problem has been chosen by a workshop of
InternationalAssociation for Hydraulic Research (IAHR) working
group (Napolitanoet al., 1985) (15) as a suitable test case for
assessing the capabilities of thecurrent numerical methods on
refined modeling of the flows on the subjectof computing laminar
flows in complex geometry. The total length of thechannel is chosen
to be Re/3. The lower boundary (solid wall) of thechannel is given
by the following expression:
yl(x)=125tanh 1230 x
Re2 tanh(2)6 (11)
On Implementation of Boundary Conditions 547
-
At the inlet, the velocity profile is given as
u=1.5(2yy2)v=0
for x=0, 0 [ y [ 1 (12)
The velocity boundary condition produced by Zou and He (14) is
used hereat the inlet.
The mesh size used for this simulation is 7131. Figures 7 and 8
showthe wall vorticity distribution for Re=10 and Re=100. The
present resultscompare well with the benchmark solution of IAHR
workshop (Napolitanoet al., 1985) given by Cliffe et al. using a
finite element method with resultsbeing grid-independent. Figure 9
displays the wall vorticity distribution fordifferent Reynolds
numbers calculated by the present method. It is con-firmed in this
figure that as Reynolds number increases to the value ofmuch larger
than 1, the solution takes on a quasi-self-similar form, i.e.,
thewall vorticity becomes independent of Re when plotted vs.
x/xout. Figure 10shows the streamlines for Re=10. The separation
region is shown clearly inthis figure.
The above numerical results show that the new boundary method
isvalid for solving the flow problem with complex geometry of
boundaries.
3.3. Special Treatment on the Corner Points
For some flow problems, which have the corner points such as
drivencavity problem, the implementation of the boundary condition
at the
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1 1.2
x/xout
w
present
result of Cliffe
Fig. 7. Wall vorticity distribution for expansion channel flow
at Re=10.
548 Chew et al.
-
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1 1.2
x/xout
w
present
result of Cliffe
Fig. 8. Wall vorticity distribution for expansion channel flow
at Re=100.
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
0 0.2 0.4 0.6 0.8 1 1.2
x/xout
w
Re>100
Re=100
Re=10
Fig. 9. Wall vorticity distribution for expansion channel flow
at different Reynolds numbers.
Fig. 10. Streamline for expansion channel flow at Re=10.
On Implementation of Boundary Conditions 549
-
corner points is very important. At the corner points, the
velocity direc-tions of the 2-D model are show in Fig. 11. For the
direction 6 and direc-tion 8, the values for these two directions
have little influence on the resultsof the numerical simulation
using the original LBM, because they do notcontribute any
information into the interior parts. But for finite volumeLBM,
these values will be used when calculating the interior points at
thesetwo directions. So it is very crucial for the choice of the
values at these twodirections. Take the case of driven cavity
problem as an example. AtRe=100, the values of the equilibrium
function were given for the distri-bution functions at these two
directions at the beginning. The primary andthe right bottom
vortices can be captured correctly and the left bottomsmall vortex
cannot be obtained. This means that the equilibrium
boundarycondition does not have enough accuracy for the complex
flow pattern.The left bottom small vortex is plagued by the
numerical errors. In order toincrease the accuracy, the first order
extrapolation scheme at these twodirections was tried. At these two
directions, the values are obtained byextrapolation from the two
interior points, and for other directions, thehalf-covolume
combined with the bounce back method are used. Thenumerical results
of this treatment are very accurate. The results forRe=100 are
shown in Figs. 1215. The mesh size used for these results
is101101.
According to the present study, for Re=100, the center of the
primaryvortex is at x=0.617, y=0.737, the center of the left corner
vortex is atx=0.030, y=0.037, and the center of the right corner
vortex is at x=0.0945, y=0.060. The results are in good agreement
with those by Ghiaet al. (16) ( for Re=100 by Ghia et al.,
x=0.6172, y=0.7344 for the primaryvortex, x=0.0313, y=0.0391 for
the left corner vortex, and x=0.9453,y=0.0625 for the right corner
vortex.).
7
1
5
8
2 6
3
4
x
y
Fig. 11. Schematic plot of velocity directions of 2-D model at
the left-bottom conner point.
550 Chew et al.
-
Fig. 12. Streamlines for driven cavity flow at Re=100.
5
4
3
2
1
0
0
-1
-2
-3
-4
-5
0.5
-0.5
-0.5
-1
-2
-3
-4
-5
Fig. 13. Vorticity contours for driven cavity flow at
Re=100.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1
u-velocity
y
present
results of Ghia
Fig. 14. Comparison of u-velocity profiles along the vertical
centerline of driven cavity atRe=100.
On Implementation of Boundary Conditions 551
-
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 0.2 0.4 0.6 0.8 1
x
v-v
eloctiy
present
results of Ghia
Fig. 15. Comparison of v-velocity profiles along the horizontal
centerline of driven cavity atRe=100.
To further validate the present method, numerical computations
wereperformed for Re=400 and Re=1000, and the results are shown
inFigs. 1619. The mesh sizes used for Re=400 and Re=1000 are
respec-tively 201201 and 251251. Clearly, a much larger number of
meshpoints is needed for Re=400 and 1000 in order to obtain
accurate numeri-cal results. This may be one of the shortcomings of
the finite volume LBMfor simulation of flows with high Reynolds
number. In this paper, we aimto improve the implementation of
boundary conditions rather than thefinite volume LBM itself. Our
numerical results for the driven cavityproblem showed that the
special treatment on the corner points is neededand effective.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-0.5 0 0.5 1
u-velocity
y
present
result of
Ghia
Fig. 16. Comparison of u-velocity profiles along the vertical
centerline of driven cavity atRe=400.
552 Chew et al.
-
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0 0.5 1 1.5
x
v-v
elocity
present
result of
Ghia
Fig. 17. Comparison of v-velocity profiles along the horizontal
centerline of driven cavity atRe=400.
0
0.2
0.4
0.6
0.8
1
1.2
-0.5 0 0.5 1 1.5
u-velocity
y present
result of
Ghia
Fig. 18. Comparison of u-velocity profiles along the vertical
centerline of driven cavity atRe=1000.
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0 0.2 0.4 0.6 0.8 1 1.2
x
v-velocity
present
result of
Ghia
Fig. 19. Comparison of v-velocity profiles along the horizontal
centerline of driven cavity atRe=1000.
On Implementation of Boundary Conditions 553
-
3.4. The Fourth Order RungeKutta Scheme
In order to accelerate the calculation speed, the fourth order
RungeKutta scheme is used in this work. The general mth order
RungeKuttascheme can be written as
f (0)=f (n) (13a)
f (k)=f (0)+ak Df (k1), k=1,..., m (13b)
f (n+1)=f (m) (13c)
where
Df (k1)=DtsP1 Caround P
collisions(f(k1)) Caround P
fluxes(f (k1))2When m is taken as 4, we can get the fourth order
RungeKutta scheme, inwhich the coefficients are taken as
a1=0.0833, a2=0.2069, a3=0.4265, a4=1
To test the efficiency of the RungeKutta scheme, we
performednumerical integration in the time direction by two
schemes: Euler explicitscheme and the fourth order RungeKutta
scheme. Two cases were studied.The first case is the expansion
channel flow. The Reynolds number is taken as10 and the mesh size
is 7131. The convergence criterion is set to
Ci, j|`(u2i, j+v2i, j)n+1`(u2i, j+v2i, j)n|;C
i, j|`(u2i, j+v2i, j)n| [ 106 (14)
where n is the time level. It was found that the time step for
the fourthorder RungeKutta scheme can be taken as 6.5 times greater
than that forthe Euler explicit scheme. The second case is the
driven cavity flow. TheReynolds number is taken as 100 and the mesh
size is chosen as 101101.The convergence criterion is set the same
as in the first case. For this case,the time step of the fourth
order RungeKutta scheme can be 5 times largerthan that of the Euler
explicit scheme. For both cases, the overall CPUtime required by
the fourth order RungeKutta scheme on Compaq ES40workstation is
less than that by the Euler explicit scheme. But the Eulerexplicit
scheme requires less virtual storage as compared to the fourth
orderRungeKutta scheme. The CPU time (s) and the memory required by
Eulerexplicit scheme and the fourth order RungeKutta scheme for
simulationof expansion channel flow and the lid driven cavity flow
are shown inTable I.
554 Chew et al.
-
Table I. Comparison of CPU Time and Memory Between Euler
Explicit Scheme andFourth Order RungeKutta Scheme
Euler explicit 4th order RungeKutta
CPU time Memory CPU time MemoryProblem Mesh size (seconds) (MB)
(seconds) (MB)
Expansion channel flow 7131 485.10 11 336.25 12Driven cavity
flow 101101 1658.51 15 1219.42 19
4. CONCLUSION
A new implementation of boundary conditions for the finite
volumeLBM has been developed in this paper. It is based on the
half-covolumeand the bounce-back rule for the non-equilibrium
distribution function.For the test problems of expansion channel
flow and driven cavity flow,good results can be obtained using this
new approach.
The fourth order RungeKutta integration is found to be a
practicalway in the LBM to enlarge the time step, so that the
convergence rate canbe speeded up.
ACKNOWLEDGMENTS
The authors thank Dr. Xiaoyi He for his useful and instructive
dis-cussion during this work.
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1. INTRODUCTION2. THE FINITE-VOLUME LBM MODEL3. IMPLEMENTATION
OF BOUNDARY CONDITIONS4. CONCLUSIONACKNOWLEDGMENTS